Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$.  Find $g^{-1}(f(-7))$.
Solution: We know that $f^{-1}(u)=v$ is the same as $u=f(v)$.  Therefore $f^{-1}(g(x))=5x+3$ is the same as  \[g(x)=f(5x+3).\]We can also use that $g(s)=t$ is equivalent to $s=g^{-1}(t)$ to say \[x=g^{-1}(f(5x+3)).\]This gives an expression containing $g^{-1}\circ f$.

Now we solve: \[g^{-1}(f(-7))=g^{-1}(f(5(-2)+3)).\]If $x=-2$ the equation $g^{-1}(f(5x+3))=x$ tells us  \[g^{-1}(f(5(-2)+3))=\boxed{-2}.\]